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Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Martínez Alonso–Shabat equation

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Abstract

We study a Lagrangian extension of the 5d Martínez Alonso–Shabat equation \(\mathcal {E}\)

$$\begin{aligned} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{aligned}$$

that coincides with the cotangent equation \(\mathcal {T}^{*}\mathcal {E}\) to the latter. We describe the Lie algebra structure of its symmetries (which happens to be quite nontrivial and is described in terms of deformations) and construct two families of recursion operators for symmetries. Each family depends on two parameters. We prove that all the operators from the first family are hereditary, but not compatible in the sense of the Nijenhuis bracket. We also construct two new parametric Lax pairs that depend on higher-order derivatives of the unknown functions.

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Notes

  1. Since \(\mathcal {C}\) is horizontal and n-dimensional, a flat connection in \(\pi _\infty \) is associated to this distribution (the Cartan connection).

  2. For a full description of the deformations of the subalgebra \(\mathbb {R}_n[\tau ] \otimes {{\,\textrm{Der}\,}}(\mathbb {R}[s]) \subset \mathfrak {q}_{n,0}\) see [45].

  3. Most of these formulas were obtained in [4].

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Acknowledgements

Computations were supported by the Jets software, [2].

Funding

The work of ISK was partially supported by RSF Grant 21-71-20034.

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  1. Trapeznikov Institute of Control Sciences, 65 Profsoyuznaya Street, Moscow, 117997, Russia

    I. S. Krasil’shchik & O. I. Morozov

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  1. I. S. Krasil’shchik
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  2. O. I. Morozov
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Krasil’shchik, I.S., Morozov, O.I. Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Martínez Alonso–Shabat equation. Anal.Math.Phys. 13, 2 (2023). https://doi.org/10.1007/s13324-022-00763-w

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